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font-align:font-size:100%;font-size: 1.4em"> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}b>a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>b</mi> <mo>></mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}b>a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d1a11fb9ebad61038e799bcb9b74b2fb5c32dab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle {}b>a}"></span>, and suppose that </p> <dl><dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g\colon [a,b]\longrightarrow \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo stretchy="false">⟶</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,g\colon [a,b]\longrightarrow \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62d73c59b59bcbf77e12fcb056dfc8687ee91d08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.791ex; height:2.843ex;" alt="{\displaystyle f,g\colon [a,b]\longrightarrow \mathbb {R} }"></span> </dd></dl> <p>are <a href="/wiki/Real_function/Continuity_in_a_Point/General/Definition" title="Real function/Continuity in a Point/General/Definition">continuous</a> functions which are <a href="/wiki/Differentiable_function/D_in_R/Via_limit/Definition" title="Differentiable function/D in R/Via limit/Definition">differentiable</a> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}]a,b[}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo stretchy="false">]</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">[</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}]a,b[}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/270d5e0116fb92c1667fca123c454c114a042fd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle {}]a,b[}"></span> and such that </p> <dl><dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}g'(x)\neq 0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}g'(x)\neq 0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34e2b9a4627a6ef2db761ce0ef6d84a616de9b85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.59ex; height:3.009ex;" alt="{\displaystyle {}g'(x)\neq 0\,}"></span></dd></dl> <p>for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}x\in {]a,b[}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">]</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">[</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}x\in {]a,b[}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50d5258fad15c7c5b7180a8778c8e887d86864ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.725ex; height:2.843ex;" alt="{\displaystyle {}x\in {]a,b[}}"></span>. </p> <div style="font-weight:bold"> Then <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}g(b)\neq g(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}g(b)\neq g(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a07082eaffff53fcf7ff1e5db2394faaf1caa29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.176ex; height:2.843ex;" alt="{\displaystyle {}g(b)\neq g(a)}"></span>, and there exists some <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}c\in {]a,b[}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>c</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">]</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">[</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}c\in {]a,b[}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/893bc6fb00faaeb9f8bf828b68e80a64823b1d6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.402ex; height:2.843ex;" alt="{\displaystyle {}c\in {]a,b[}}"></span> such that </p> <dl><dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}{\frac {f(b)-f(a)}{g(b)-g(a)}}={\frac {f'(c)}{g'(c)}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}{\frac {f(b)-f(a)}{g(b)-g(a)}}={\frac {f'(c)}{g'(c)}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21c29462e120a846f1966f08dabbc942cae80209" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.87ex; height:6.509ex;" alt="{\displaystyle {}{\frac {f(b)-f(a)}{g(b)-g(a)}}={\frac {f'(c)}{g'(c)}}\,.}"></span></dd></dl> </div> </div> <div class="center"><a href="/wiki/Differentiable_function/Mean_value_theorem/Quotient_version/Fact/Proof" title="Differentiable function/Mean value theorem/Quotient version/Fact/Proof">Proof</a>, <a class="external text" 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